UNIVERZITA KOMENSKÉHO
(English Version)
Il Concetto di variabile nel passaggio dal linguaggio Aritmetico al linguaggio Algebrico
(Versione Italiana)
Referee:
1. Prof. RNDr. Jozef Fulier, PhD.
2. Doc. RNDr. Vladislav Rosa, PhD.
3. PaedDr. Michaela Regecová, PhD.
5. Prof. Filippo Spagnolo, PhD (Supervisor)
STORY OF THE PRECEDING WORKS
The aim of the experimental research effected in Malisani (1990, 1992) was to study the cognitive performance of the students between the ages of 14-15 in the assignment of resolution of algebraic and geometric problems. We wanted to know more specifically how the different kind of logical structure of a problem affect the resolving performance (types of solutions, steps of the resolving algorithm and errors); and if it is verified that the isomorphism of logical structures in the algebraic and geometric contexts does not implicate isomorphism in the performance of the students.
The problems belonging to the algebraic context are concerned with the resolution of equations of first degree with one unknown, of the type: y = k1 . (x - k2 ) [1] for a determined value of x or y, being k1 and k2 positive constants and such that k2 < x. In this type of equation the variables x and y and the constants k1 and k2 can represent any elements, therefore, they have only a formal significance.
The geometric problems consider, instead, the application of the theorem of the sum of the interior angles of a convex polygon that has equation: s = 180°. (n - 2) [2], in which every variable and every constant represent determined geometric objects or relations among these objects. For example: s is the sum of the interior angles of a convex polygon, 180° is the sum of the interior angles of a triangle, n is the number of sides of a polygon, n 2 is the number of triangles that are determined in the polygon tracing the diagonals from a vertex to the others. In this case the variables and the constants have a geometric significance.
We observe that the equations [1] and [2] are isomorphic with regard to their logical structure, because if we fix a variable (for example: y and s) they require the same steps for their resolution. These equations are of arithmetical kind, using the terminology of Gallardo and Rojano (1988), because to resolve them it is necessary to manipulate only the numerical values of the equation (actions in the arithmetic context) and not the quantities to find or unknowns.
From the results obtained we deduce that the geometric significance of a problem:
Ø would affect partially the achievement of correct answers, only in those problems that introduce greater logical difficulty (7 or more different steps).
Ø influences positively the economy of steps of the resolution.
Ø affects the number and the kind of errors made by the pupils.
Therefore, the intuitive support that the geometric problems offer and a good comprehension by the pupils of the conceptual relations between the elements that compose the equation of the theorem favour the saving of steps in the resolution and decrease the number of errors, above all the errors of calculation. That is, the saving of steps in the resolution does not always implicate a greater quantity of errors, contrary to what is usually supposed.
At that time the formulated conclusions affirmed that the resolution of problems that involves equations requires something more than the domain of certain operations (arithmetical and algebraic); the subjects must have the necessary conceptual knowledge to understand and to represent conveniently the information of the problem.
The individualization and the diagnosis of the errors effected in Malisani (1990) and Malisani (1992) led us to deepen the principal works of research carried out in these last decades on the cognitive processes associated with the learning of algebra (Matz, 1982; Kieran & Filloy, 1989; Kieran, 1991; Gallardo & Rojano, 1988; Lee & Wheeler, 1989; Chiappini and Lemut, 1991; Herscovics & Linchevski, 1991). These studies deal with matters concerning the difficulties and obstacles that the beginner students of algebra meet, regarding the conceptual changes necessary in the transition from the arithmetic thought to the algebraic thought. These changes refer especially to the concept of equality, the conventions of notation and the interpretation of the concept of variable. We also examined the results of some researches on the interpretation and simplification of algebraic expressions and the resolution of equations and algebraic problems (Malisani, 1993).
Successively we carried out a research on the individualization, diagnosis and classification of errors in the resolution of algebraic and geometric problems that involve arithmetical equations of first degree (Malisani, 1993). Even if the resolving procedure of the algebraic and geometric problems is isomorphic two by two, the results we obtained point out that the students do not make the same types of errors. For example, the percentages of errors related to the use of the equal sign and to the transport of terms from a member to the other of the equation are lower in the geometric context. On the other hand, the percentage of errors concerning the formulation of an answer consistent with the meaning of the variables that represent the results is lower in the algebraic context.
Several experimental studies (Harper, 1987; Sfard 1992) seem to confirm that some difficulties of the students can be grouped around some obstacles met in history (Cfr. Arzarello, pp. 7-8). The elements that allow to identify these obstacles have to be searched in the analysis of the resistances emerged in the historical development and in the debates that have overcome them. But history alone is not sufficient, the historical epistemological analysis must be completed by a study of the grounding of mathematics (Spagnolo, 1995, pp. 18-19). If we consider this point of view, it could be useful to take into consideration the history of the algebraic thought that leads us to go over the steps of the construction of the algebraic language.
The historical analysis effected in Malisani (1996, 1999) shows that for many centuries algebra stayed behind in comparison with the geometry and that the construction of the symbolic language was too slow and difficult. The lack of an adequate algebraic language conditioned the evolution of the resolutive procedures. The ancient mathematicians often explained these procedures through their application to some examples. They used other languages: natural, arithmetical and geometric.
PURPOSE OF THE RESEARCH
To deepen the conclusions previously expressed a new research is proposed. It is founded on the necessity of studying and analyzing the obstacles that the students meet in building up and assimilating certain concepts, in the passage from the arithmetical thought to the algebraic thought.
From some effected studies (Matz, 1982; Wagner, 1981, 1983) it emerges that the point of critical transition between the two kinds of thought is the introduction of the concept of variable. This notion could take on a plurality of conceptions: generalized number (2+4 = 4+2 is generalized with a+b = b+a); unknown (resolution of equations); something that varies (relation among quantities, functional aspect); entirely arbitrary sign (study of the structures); register of memory (in computer science) (Usiskin, 1988).
The study of the various aspects that this concept can take constitutes a very wide field of research and requires different confirmations, provided by historical-epistemological and experimental investigation and by setting up the didactical situations built ad-hoc. Therefore it is necessary to circumscribe the dominion of survey.
The aim of this research is to study some characteristics of the period of transition from the arithmetical language to the algebraic language. We want to analyze if the different conceptions of variable are evoked by the students in the resolution of problems and if the notion of variable in its double aspect unknown and relational-functional represents an obstacle for the pupil.
APPLICATIONS
This research is set as a contribution to Mathematics Education, particularly, to the studies that are being carried out within the GRIM, on the epistemological and didactical obstacles concerning the passage from the arithmetic language to the algebraic language.
This experimental study will supply us some necessary tools to analyze in details whether the concept of variable, in its different aspects, represents an epistemological obstacle or an obstacle of didactical origin.
We could also determine how the semiotic context influences the conceptions of variable from the pupils point of view. We could study more specifically the interaction of other contexts natural language, geometric language, perceptive schemes, etc. with the operating of the pupils in a strictly algebraic context.
Moreover, it will be possible to draw some tools to set up appropriate a-didactical situations and to get at a more deep comprehension of the communicative processes.
From a general point of view, this research can help to clarify matters concerning the representations of the arithmetical and algebraic knowledge and the operating in the resolution of problems from the pupils point of view.
STRUCTURE OF THE THESIS
The thesis is composed by five chapters. The first one is about history and introduces the construction of the algebraic language and the evolution of the methods and of the strategies of resolution of equations in the periods that preceded the formalization.
The second chapter has the purpose to study some aspects of the period of transition from the arithmetical language to the algebraic language. We want to analyze if the pupils evoke the different conceptions of variable in the resolution of problems and if the natural language and/or the arithmetic language prevail as symbolic systems in absence of an adequate mastery of the algebraic language.
The third chapter intends to study the relational-functional aspect of the variable in problem-solving, considering the semiotic contexts of algebra and analytical geometry. The aim is to investigate whether the notion of unknown interferes with the interpretation of the functional aspect, and whether the procedures in natural language and/or the arithmetical language prevail as resolutive strategies in lack of an adequate knowledge of the algebraic language.
The aim of the fourth chapter is to analyze how the conceptions of unknown and of functional relation are activated and used in the process of resolution of a problematic situation. We want also to study the process of translation from the algebraic language into the natural one and the representation of the syntax-semantics relation within the algebraic code.
In the fifth chapter the final conclusions of the thesis are presented.
INDEX
Story of the preceding works . 5
Purpose of the research ...7
Applications .8
Structure of the thesis . 8
Bibliography .. 9
1.1 Introduction . 11
1.2 The symbolism 12
1.3 Methods of resolution of equations . 17
1.3.1. Methods of resolution of the equations of first degree ... ... 17
1.3.1.1. The geometric procedure of Euclid 17
1.3.1.2. The methods of the false position . 17
1.3.1.2.1. The methods of the simple false position . 18
1.3.1.2.2. The methods of the double false position . 19
1.3.1.3. The regula infusa . 21
1.3.2. Methods of resolution of the equations of second degree . ....... 23
1.3.2.1. The geometric procedure of Euclid . . 23
1.3.2.2. The procedure of al-Khowârizmî 24
1.3.2.3. The cut-and-paste geometry . 25
1.3.3. Methods of resolution of the equations of third degree 26
1.3.3.1. The procedure of al-Khayyam 26
1.3.3.2. The procedure of al-Tusi . 28
1.3.4 Methods of resolution of the indeterminate equations .. 29
1.3.4.1 The procedure of Diophantus .. 29
1.3.4.2 The method of pulverization . ... 29
1.3.4.3 The procedure of Abu Kamil 31
1.3.5. European methods up to 1500 .. 31
1.3.6 Conclusions on the methods of resolutions 33
1.4 The negative numbers as obstacle. The incomplete numerical field .. 35
1.5 Generalization of the problems .. 37
1.6 The variable as thing that varies .. 38
1.7 Conclusions .....40
Notes 42
Bibliography . 44
2.1. Introduction 47
2.2. Hypothesis .. 49
2.3. Experimental reports for scholastic level .. 49
2.3.1 Middle School .... 49
2.3.1.1 The a-didactic situation and its phases .. 49
2.3.1.2 The a-priori analysis .. 50
2.3.1.3 Quantitative analysis of the data 51
2.3.1.4 Qualitative analysis ... 53
2.3.1.5 Discussion of the results 55
2.3.2 High School 56
2.3.2.1 The a-didactic situation and its phases .. 56
2.3.2.2 Description of the phase of validation .. 58
2.3.2.3 The a-priori analysis .. 58
2.3.2.4 Quantitative analysis of the data 60
2.3.2.5 Qualitative analysis 62
2.3.2.6 Discussion of the results 63
2.4 Conclusions 64
Bibliography ..65
Appendix N° 1 .. 67
Appendix N° 2 .. 68
Appendix N° 3 .. 69
Appendix N° 4 . 71
Appendix N° 5 . 73
Appendix N° 6 . . 74
Chapter 3. The notion of variable in different semiotic contexts 83
3.1. Introduction ... 83
3.2 Methodology of the research 84
3.3 A- priori analysis 86
3.4 The hypotheses and the a-priori table 89
3.5 Quantitative analysis .. 91
3.5.1 First implicative analysis and comments of the first problem .. 92
3.5.2 Falsification of H1 . 94
3.5.3 Profile of the pupils 94
3.5.4 The hierarchical tree .. 95
3.5.5 The factorial analysis by S.P.S.S. .. 96
3.5.6 Second implicative analysis and comments of the first problem .. 97
3.5.7 Falsification of H2 98
3.5.8 Third implicative analysis and comments of the second problem 99
3.5.9 Falsification of H3 . 100
3.5.10 Fourth implicative analysis and comments of the fourth problem . 100
3.5.11 Fifth implicative analysis and comparison between the first and fourth problem 102
3.5.12 Falsification of H4 103
3.6 Conclusions . 103
Notes 105
Bibliography 106
Appendix N° 1 109
Appendix N° 2 110
Appendix N° 3 113
Appendix N° 4 115
Appendix N° 5 115
Chapter 4. The variable between unknown and thing that varies. Some aspects of the symbolic language 116
4.1. Introduction . .. 116
4.2. Methodology of the research ... 117
4.3 Analysis of the protocols of the first problem 117
4.3.1 First pair: Serena and Graziela .. .. 117
4.3.1.1 Types of language 117
4.3.1.2 Resolutive procedure .. 118
4.3.1.3 Comments 118
4.3.2 Second pair: Vita and Alessandra . 119
4.3.2.1 Types of language 119
4.3.2.2 Resolutive procedure 119
4.3.2.3 Comments 121
4.4 Analysis of the protocols of the second problem 123
4.4.1 First pair: Serena and Graziela 123
4.4.1.1 Resolutive procedure .. 123
4.4.1.2 Comments 124
4.4.2 Second pair: Vita and Alessandra . 124
4.4.2.1 Resolutive procedure .. 124
4.4.2.2 Comments ... 126
4.5 Final conclusions .. 128
Notes 131
Bibliography 131
Appendix N° 1 ..133
Appendix N° 2 . 134
Appendix N° 3 138
Chapter 5. Final conclusions 159
Bibliography 166